>> Okay, so, the Pluto problem looks like this. Here's our sun. Pluto is in this elliptical orbit about the sun. And there is some close approach, we'll call that R1. There is some distance, R2. And what they tell us is the following, R1 equals 4.43 times 10 to the 9 kilometers. R2 is something else, I'll give it to you in a second. V1 is 6.12. R2 is 7.3 times 10 to the 9 kilometers. Okay. And we want to figure out what V2 is. All right. So, let me start that again. Hello, class, Professor Anderson here, let's take a look at Pluto's orbit. Pluto has this very elliptical orbit and what we know is that at closest approach it has a distance of 4.43 times 10 to the 9 kilometers and it's moving at a speed of 6.12 kilometers per second. When it gets out to its farthest point it is moving at some speed V2. We know that distance is 7.3 times 10 to the 9 kilometers. And we need to figure out what V2 is. Okay. So, let's take a guess and then we'll see if we can calculate it. What do you think V2 is going to be? Is it going to be bigger than V1, equal to V1, smaller than V1, what do you guys think? >> Smaller. >> Smaller. Why? Who has a thought? Jamie, you want to say something. Hand the mic to Jamie. Okay, Jamie, we just took a guess that V2 would be smaller than V1, why do you think that? >> The distance is further, so the speed would have to be slower for the area to be equal to the other. >> Right. Exactly right. If Kepler's second law is going to hold, then equal areas and equal time says that V2 has to be smaller than V1. Exactly right. As a follow up, how do we calculate it? >> Umm. I'm not sure. >> Okay. Well. >> Find the area of one and make it proportional to the other. >> Okay. Let's see if we can do that. So, let's draw little areas here and let's -- let's maybe just redraw our ellipse over here. Okay, here's our ellipse, here's the sun. We've got this area, which we're going to call A1. And we've got this area, let's call it A2. We know what this distance is, R1. We know what that distance is, R2. If I knew the other sides of these triangles then I could calculate exactly what the area is. Do I know this side of the triangle? Not really, right? But if it's a very small side of the triangle here, we can approximate it. It is how fast you're moving, times how long did it take you. Yeah? That's the distance. What about over here? I could do the same thing on this side. This would be V2 DT. And what Jamie said was, "If Kepler's second law is going to hold, those DTs, those little deltas, have to be exactly the same." Right? So, if the areas are the same, than what do we have? We know the area of a triangle, it's just 1/2 the base times the height. R1 times V1 times DT. What about A2? It's 1/2 R2, V2, DT. Ah ha. Cross out the halves, cross out the DTs, I get V2 is equal to R1 over R2 times V1. And now we have all those numbers, so we can plug it in and try it. R1 is 4.43 times 10 to the 9 kilometers. I'm going to divide by 7.3 times 10 to the 9 kilometers. And I'm going to multiply by V1, which is 6.12 kilometers per second. This number is certainly less than 1. Our guess is going to hold for sure. What's the actual number? I don't know. Punch it into your calculator and tell me what you get. I will approximate it here. So, we've got 4.43 over 7.3. We're going to multiply that by 6.12. Ah, let's see, this is 4.5 over 7.5, which is -- if I divide those by 15, right, that would be 3 over 5. So, that's approximately 3/5 of 6 and 3/5 of 6 is what? That's 18 over 5, which is pretty close to 3.6. Anybody get an answer? >> 3.7. >> 3.7. All right, so our approximation was pretty good. We get 3.7 kilometers per second. All right? Certainly less than V1 and it scales just like the radii, R1 to R2. So, this answer right here tells you that if it's a circular orbit, obviously your speed is the same everywhere, because R1 is equal to R2 everywhere. And as it gets more and more elliptical this is going to get faster and that's going to get slower. Okay? Anybody feel sorry for Pluto? Used to be a planet and then got demoted. I feel a little sad. But that's what happens when you find other things in your solar system that are bigger than that. Either you make all those other things planets, or you demote Pluto to a non-planet, which is what happened, right?